The field of the invention is nuclear magnetic resonance imaging methods and systems. The invention relates to the correction of image artifacts caused by "Maxwell terms" arising from imaging gradients in MRI systems, and more particularly, to the correction of Maxwell phase errors produced by non-rectilinear k-space sampling methods such as spiral scanning, projection imaging and twisted radial line imaging.
When a substance such as human tissue is subjected to a uniform magnetic field (polarizing field B.sub.0), the individual magnetic moments of the spins in the tissue attempt to align with this polarizing field, but precess about it at their characteristic Larmor frequency. If the substance, or tissue, is subjected to a magnetic field (excitation field B.sub.1) which is in the x-y plane and which is near the Larmor frequency, the net aligned moment, M.sub.z, may be rotated, or "tipped", into the x-y plane to produce a net transverse magnetic moment M.sub.t. A signal is emitted by the excited spins, and after the excitation signal B.sub.1 is terminated, this signal may be received and processed to form an image.
When utilizing these signals to produce images, magnetic field gradients (G.sub.x, G.sub.y, and G.sub.z) are employed. Typically, the region to be imaged is scanned by a sequence of measurement cycles in which these gradients vary according to the particular localization method being used. The resulting set of received NMR signals are digitized and processed to reconstruct the image using one of many well known reconstruction techniques.
It is well known that imperfections in the linear magnetic field gradients (G.sub.x, G.sub.y, and G.sub.z) produce artifacts in the reconstructed images. It is a well known problem, for example, that eddy currents produced by gradient pulses will distort the magnetic field and produce image artifacts. Methods for compensating for such eddy current errors are also well known as disclosed, for example, in U.S. Pat. Nos. 4,698,591; 4,950,994; and 5,226,418.
It is also well known that the gradients may not be perfectly uniform over the entire imaging volume, which may lead to image distortion. Methods for compensating this non-uniformity are well known, and for example, are described in U.S. Pat. No. 4,591,789.
Other than uncompensated eddy current errors and gradient non-uniformity errors that escape correction, it can be assumed that the magnetic field gradients (G.sub.x, G.sub.y, and G.sub.z) produce linear magnetic fields exactly as programmed, thus spatially encoding the NMR data accurately. With these gradients, the overall static magnetic field at location (x,y,z) is conventionally given as B.sub.0 +G.sub.x x+G.sub.y Y+G.sub.z z, and the direction of the field is usually thought to be along the z-axis. This description, however, is not exactly correct. As long as a linear magnetic field gradient is applied, the overall magnetic field is nutated away from the z-axis and its amplitude exhibits higher-order spatial dependencies (x.sup.2, y.sup.2, z.sup.2, z.sup.3, . . . ). These phenomena are a direct consequence of the Maxwell equations which require that the overall magnetic field satisfy the following two condition: .gradient..multidot.B=0 and .gradient..times.B=0. The higher-order magnetic fields, referred to as "Maxwell terms" (or Maxwell fields), represent a fundamental physics effect, and are not related to eddy currents or imperfection in hardware design and manufacture. Although Maxwell terms have been known for at least a decade, their effect on imaging has been largely ignored because of their minute amplitudes under conventional imaging conditions.
MR imaging methods may be categorized by the k-space trajectory used. Conventional, "rectilinear" MR imaging methods traverse k-space using a set of parallel lines which sample a rectangle in 2D or a parallelpiped (box) in 3 D space. MR imaging methods which use substantially non-rectilinear k-space sampling include spiral scans, projection imaging, circular scans and twisted radial line imaging. One main difference between rectilinear and non-rectilinear sampling patterns is the type of artifacts which result. For substantially rectilinear k-space trajectories, the Maxwell fields cause ghosting and geometrical distortion but with non-rectilinear trajectories, blurring results. Correction methods for removing ghosting and geometrical distortion from the substantially rectilinear trajectories are available, but these methods are not appropriate for non-rectilinear trajectories because of the different nature of the artifacts.